Chapter 18: Temperature, Heat, and the First Law of Thermodynamics

Temperature

Section 18.1 Temperature

Learning Objectives

• Lowest temperature is 0 on the Kelvin scale (absolute zero).

• Zeroth law of thermodynamics.

• Conditions for the triple-point temperature.

• Conditions for measuring temperature with a constant-volume gas thermometer.

• Relate pressure and temperature of a gas in a given state to the pressure and temperature at the triple point for a constant-volume gas thermometer.

Definition of Temperature

• Thermodynamics studies thermal energy (internal energy) of systems; temperature is a central concept.

• Temperature is an SI base quantity related to hot and cold, measured with a thermometer.

• Thermometers use a working substance with a measurable property (length, pressure) that changes with temperature.

• Physicists use the Kelvin scale (units: kelvins).

Zeroth Law of Thermodynamics

• Two bodies are in thermal equilibrium if they have the same temperature throughout, and no heat flows between them.

• If bodies A and B are each in thermal equilibrium with a third body T, then A and B are in thermal equilibrium with each other.

Triple Point of Water

• The triple point of water is where solid ice, liquid water, and water vapor coexist in thermal equilibrium (not at normal atmospheric pressure).

• The temperature of this mixture is defined as $273.16 K$.

• A constant-volume gas thermometer's bulb is inserted into the well of a triple-point cell.

Constant-Volume Gas Thermometer

• A gas-filled bulb connects to a mercury manometer via a tube.

• The mercury level in the U-tube's left arm is kept at zero to maintain constant gas volume.

• The temperature is measured using the equation: $(T = (lim<em>{p \to 0})\frac{p}{p</em>3} (273.16 K))$, where p is the observed pressure and $p_3$ is the pressure at the triple point of water.

Section 18.2 The Celsius and Fahrenheit Scales

Learning Objectives

• Convert temperature between Celsius, Fahrenheit, and Kelvin scales.

• A change of one degree is the same on the Celsius and Kelvin scales.

Temperature Conversion Equations

• Celsius scale definition: $(T_C = T - 273.15)$, where T is in kelvins.

• Fahrenheit scale definition: $(T<em>F = \frac{9}{5}T</em>C + 32)$.

Celsius and Fahrenheit Reference Values

Section 18.3 Thermal Expansion

Learning Objectives

• Apply the relationship between temperature change $(\Delta T)$, length change $(\Delta L)$, initial length L, and the coefficient of linear expansion $\alpha$ for one-dimensional thermal expansion.

• Use one-dimensional thermal expansion to find the change in area for two-dimensional thermal expansion.

• Apply the relationship between temperature change $(\Delta T)$, volume change $(\Delta V)$, initial volume V, and the coefficient of volume expansion $\beta$ for three-dimensional thermal expansion.

Definition of Linear Thermal Expansion

• Objects change size with temperature changes. The change in any linear dimension L for a temperature change $\Delta T$ is given by: $\Delta L = L \alpha \Delta T$, where $\alpha$ is the coefficient of linear expansion.

Bimetallic Strip Temperature Sensor

• A bimetallic strip bends with temperature changes, operating thermostats by making or breaking electrical contact.

Volume Expansion Definition

• If the temperature of a solid or liquid with volume V increases by $\Delta T$, the increase in volume is: $\Delta V = V \beta \Delta T$, where $\beta$ is the coefficient of volume expansion.

• Relationship between $\beta$ and $\alpha$: $\beta = 3\alpha$.

Thermal Expansion Checkpoint #2

• Ranking rectangular metal plates with sides of L, 2L, or 3L (same material, same temperature increase):

  • Increase in vertical heights: 2 and 3 (same), then 1, then 4.

  • Increase in areas: 3, then 2, then 1 and 4 (same).

Section 18.4 Absorption of Heat

Learning Objectives

1. Thermal energy is associated with random motions of microscopic bodies in an object.

2. Heat Q is the energy transferred to or from an object's thermal energy because of a temperature difference.

3. Convert energy units between measurement systems.

4. Convert between mechanical or electrical energy and thermal energy.

5. Relate the temperature change $\Delta T$ of a substance to the heat transfer Q and the substance's heat capacity C.

6. Relate the temperature change $\Delta T$ of a substance to the heat transfer Q, the substance's specific heat c, and mass m.

7. Identify the three phases of matter.

8. Relate the heat transfer Q, the heat of transformation L, and the mass m transformed during a phase change.

9. Calculate heat transfer in steps when a substance crosses a phase-change temperature: (a) temperature change to reach the phase-change temperature, (b) the phase change, and then (c) any temperature change away from the phase-change temperature.

Temperature and Heat

• Heat Q is energy transferred between a system and its environment due to a temperature difference.

• Measured in joules (J), calories (cal), kilocalories (Cal or kcal), or British thermal units (Btu).

• Conversion: $1 cal = 3.968 \times 10^{-3} Btu = 4.1868 J$.

Heat Absorption by Solids and Liquids

• Heat capacity C: $(Q = C \Delta T = C(T<em>f - T</em>i))$, where Q is the heat absorbed or lost, and $T<em>i$ and $T</em>f$ are initial and final temperatures.

• Specific heat c: $(Q = cm \Delta T = cm(T<em>f - T</em>i))$, where m is the mass of the object.

Absorption of Heat Checkpoint #3

• Material A warms 1 g by 3°C with heat Q, and material B warms 1 g by 4°C with the same heat Q. Material A has greater specific heat (inversely proportional to the change in temperature).

Heat of Transformation and Molar Specific Heat

• Molar specific heats involve moles rather than mass units.

• Heat of transformation L: the energy per unit mass transferred during a phase change.

• Total energy transferred during a phase change: $(Q = Lm)$.

Table 18.4.1 Molar Specific Heats of Common Substances

Section 18.5 The First Law of Thermodynamics

Learning Objectives

1. Calculate the work W done by an enclosed gas as it expands or contracts by integrating the gas pressure with respect to the volume of the enclosure.

2. Identify the algebraic sign of work W associated with expansion and contraction of a gas.

3. Given a p-V graph of pressure versus volume for a process, identify the starting and final states and calculate the work by using graphical integration.

4. On a p-V graph, identify the algebraic sign of the work associated with a right-going process and a left-going process.

5. Apply the first law of thermodynamics to relate the change in the internal energy $\Delta E_{int}$ of a gas, the energy Q transferred as heat to or from the gas, and the work W done on or by the gas.

6. Identify the algebraic sign of a heat transfer Q that is associated with a transfer to a gas and a transfer from the gas.

7. Identify that the internal energy $\Delta E_{int}$ of a gas tends to increase if the heat transfer is to the gas, and it tends to decrease if the gas does work on its environment.

8. Identify that in an adiabatic process with a gas, there is no heat transfer Q with the environment.

9. Identify that in a constant-volume process with a gas, there is no work W done by the gas.

10. Identify that in a cyclical process with a gas, there is no net change in the internal energy $\Delta E_{int}$.

11. Identify that in a free expansion with a gas, the heat transfer Q, work done W, and change in internal energy $\Delta E_{int}$ are each zero.

Work to Change Volume Against a Pressure

• Gas exchanges energy with surroundings through work.

• Work W done by a gas expanding/contracting from initial volume $V<em>i$ to final volume $V</em>f$: $(W = \int<em>{V</em>i}^{V_f} p dV)$.

• Integration is needed because pressure p may vary during the volume change.

Heat Flow vs. Work for Gas Confined in a Cylinder Expanding and Contracting

• [Illustration of gas expansion and contraction]

The First Law of Thermodynamics

• Conservation of energy for a thermodynamic process: $\Delta E<em>{int} = E</em>{int,f} - E_{int,i} = Q - W$ (First Law).

• For a differential change: $dE_{int} = dQ - dW$ (First Law).

Statement of the First Law of Thermodynamics

• The internal energy $E_{int}$ of a system increases if energy is added as heat Q and decreases if energy is lost as work W done by the system.

Four Special Cases for the First Law of Thermodynamics

Section 18.6 Heat Transfer Mechanisms

Learning Objectives

1. Apply the relationship between the energy-transfer rate $P_{cond}$ and the layer's area A, thermal conductivity k, thickness L, and temperature difference $\Delta T$ for thermal conduction through a layer.

2. For a composite slab (two or more layers) that has reached the steady state, identify that the rates of thermal conduction $P_{cond}$ through the layers must be equal.

3. Apply the relationship between thermal resistance R, thickness L, and thermal conductivity k for thermal conduction through a layer.

4. Identify that thermal energy can be transferred by convection, in which a warmer fluid (gas or liquid) tends to rise in a cooler fluid.

5. Apply the relationship between the energy-transfer rate $P_{rad}$ and the object's surface area A, emissivity $\epsilon$, and surface temperature T (in kelvins) in the emission of thermal radiation.

6. Apply the relationship between the energy-transfer rate $P_{abs}$ and the object's surface area A and emissivity $\epsilon$, and the environmental temperature T (in kelvins) in the absorption of thermal radiation.

7. Calculate the net energy transfer rate $P_{net}$ of an object emitting radiation to its environment and absorbing radiation from that environment.

Thermal Conduction Definition

• Rate $P<em>{cond}$ at which energy is conducted through a slab: $(P</em>{cond} = \frac{Q}{t} = \frac{kA(T<em>H - T</em>C)}{L})$, where A is the area, L is the length, and k is the thermal conductivity.

Thermal Conduction Between Isothermal Reservoirs

• Energy transfers from a reservoir at temperature $T<em>H$ to a cooler reservoir at temperature $T</em>C$ through a conducting slab.

Convection: Practical Description

• Convection: energy transfer by motion within a fluid due to temperature differences.

• Examples: candle flame, atmospheric convection, ocean energy transfers.

Thermal Radiation Definition

• Radiation: energy transfer via electromagnetic energy emission.

• Rate $P<em>{rad}$ at which an object emits energy via thermal radiation: $(P</em>{rad} = \sigma \epsilon A T^4)$.

Thermal Radiation Equation Details

• Stefan–Boltzmann constant: $\sigma = 5.6704 \times 10^{-8} W/m^2 K^4$.

• $\epsilon$ is the emissivity, A is the surface area, and T is the surface temperature (in kelvins).

• Rate $P<em>{abs}$ at which an object absorbs energy via thermal radiation: $(P</em>{abs} = \sigma \epsilon A T<em>{env}^4)$, where $T</em>{env}$ is the environmental temperature.

Chapter 18 Summary

Temperature, Zeroth Law, and Kelvin Scale

• Temperature and Thermometer: SI base quantity related to hot and cold, measured using a thermometer.

• Zeroth Law of Thermodynamics: If A and B are each in thermal equilibrium with C, then A and B are in thermal equilibrium with each other.

• Kelvin Temperature Scale: $(T = (lim<em>{p \to 0})\frac{p}{p</em>3} (273.16 K))$.

Celsius & Fahrenheit Scales, Thermal Expansion

• Celsius Scale: $(T_C = T - 273.15)$

• Fahrenheit Scale: $(T<em>F = \frac{9}{5}T</em>C + 32)$

• Linear Expansion: $\Delta L = L \alpha \Delta T$

Volume Expansion and Heat Capacity

• Volume Expansion: $\Delta V = V \beta \Delta T$

• Heat Capacity: $Q = C(T<em>f - T</em>i)$

• Specific Heat: $Q = cm(T<em>f - T</em>i)$

First Law of Thermodynamics

• First Law: $\Delta E<em>{int} = E</em>{int,f} - E_{int,i} = Q - W$

• Differential Form: $dE_{int} = dQ - dW$

First Law Applications and Heat Transfer

• Adiabatic Processes: Q = 0, $\Delta E_{int} = -W$

• Constant-Volume Processes: W = 0, $\Delta E_{int} = Q$

• Cyclical Processes: $\Delta E_{int} = 0$, Q = W

• Free Expansions: Q = W = 0, $\Delta E_{int} = 0$

• Conduction: $P<em>{cond} = \frac{kA(T</em>H - T_C)}{L}$

• Radiation: $P_{rad} = \sigma \epsilon A T^4$